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NDA 2 2024 Maths, question 82

Topic: Algebra

Passage · Q81 to Q82

Let f∘g(x)=cos⁡2xf\circ g(x)=\cos ^{2}\sqrt{x} and g∘f(x)=∣cos⁡x∣g\circ f(x)=|\cos x|.

Q82

Which one of the following is g(x)g(x)?

  1. (a) x\sqrt{x}Correct answer
  2. (b) ∣x∣|x|
  3. (c) x2x^{2}
  4. (d) x∣x∣x|x|

Solution.

Answer: (a) x\sqrt{x}

  1. Choose g(x)=√x on x≥0 and f(x)=cos²x.
  2. Then g(f(x))=√(cos²x)=|cos x|, while f(g(x))=cos²√x.
  3. Both required compositions hold, confirming (a).